Platitude des tissus duaux de certains pré-feuilletages convexes du plan projectif complexe

A holomorphic pre-foliation $\mathcal{F}=\mathcal{C}⊠\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$ is the data of a reduced complex projective curve $\mathcal{C}$ of ${\mathbb{P}}_{\mathbb{C}}^2$ and a holomorphic foliation $\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$. When the foliation $\mathcal{F}$ is convex and the curve $\mathcal{C}$ is invariant by $\mathcal{F}$, we speak of convex pre-foliation. In a previous paper, we showed that if a foliation $\mathcal{F}$ on ${\mathbb{P}}_{\mathbb{C}}^2$ is reduced convex or homogeneous convex and if $\mathcal{C}$ is an invariant line of $\mathcal{F}$, then the dual web of the convex pre-foliation $\mathcal{C}⊠\mathcal{F}$ is flat. In this paper, we propose to extend this result to the case of a curve $\mathcal{C}$ consisting of several invariant lines.